We develop a quantum-inspired numerical procedure for searching low-energy states of a classical Hamiltonian composed of two-body fully connected random Ising interactions and a random local longitudinal magnetic field. In this method, we introduce infinitesimal quantum interactions that do not commute with the original Ising Hamiltonian and repeatedly generate and truncate direct product states, following the idea of the Krylov subspace method, to obtain the low-energy states of the original classical Ising Hamiltonian. The computational cost is controlled by the form of infinitesimal quantum interactions (e.g., one- or two-body interactions) and the numbers of infinitesimal interaction terms introduced, different initial states considered, and low-energy states kept during the iteration. For a demonstration of the method, here, we introduce pair products of Pauli X operators acting on different sites and on-site Pauli X operators into the random [sing Hamiltonian as the infinitesimal quantum interactions, in which the computational cost is O(N-3) per iteration with the system size N. We consider 120 instances of the random coupling realizations for the random Ising Hamiltonian with N up to 600 and search the 120 lowest-energy states for each instance. We find that the time-to-solution by the quantum-inspired method proposed here, with parallelization in terms of the different initial states, for searching the ground state of the random Ising Hamiltonian scales approximately as N-5 for N up to 600. We also examine the basic physical properties such as the ensemble-averaged ground-state and first-excited energies and the ensemble-averaged number of states in the low-energy region of the random Ising Hamiltonian.
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